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You are asked to sketch a graph of the piecewise function \( f(x) \) given by:

\[
f(x) =
\begin{cases}
x - 5 & \text{if } x < 1 \\
-4 + 2x & \text{if } x \geq 1
\end{cases}
\]

### Steps to sketch the graph:

1. **First part: \( f(x) = x - 5 \) when \( x < 1 \)**:
   - This is a linear function with a slope of 1 and a y-intercept of -5.
   - For \( x = 1 \), the value is \( f(1) = 1 - 5 = -4 \).
   - Since the condition is \( x < 1 \), you should draw an open circle at \( (1, -4) \) to indicate that the function doesn't include this point.

2. **Second part: \( f(x) = -4 + 2x \) when \( x \geq 1 \)**:
   - This is also a linear function with a slope of 2 and a y-intercept of -4.
   - For \( x = 1 \), the value is \( f(1) = -4 + 2(1) = -2 \).
   - Draw a closed circle at \( (1, -2) \) to indicate the function starts here, as the function is defined for \( x \geq 1 \).
   - The line continues for all \( x \geq 1 \).

### Key points for the sketch:
- Plot the line \( f(x) = x - 5 \) for \( x < 1 \), but stop just before \( x = 1 \), placing an open circle at \( (1, -4) \).
- Plot the line \( f(x) = -4 + 2x \) starting at \( (1, -2) \) with a closed circle, and extend the line for \( x > 1 \).

Would you like me to sketch this graph for you?

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Here are 5 related questions:
1. How do we calculate the slope and y-intercept of a linear function?
2. What is the significance of open and closed circles in piecewise functions?
3. Can a piecewise function be continuous at the break point, and how do we check this?
4. How does the slope affect the steepness of the lines in the graph?
5. What happens if both parts of the piecewise function have the same slope?

**Tip**: When graphing piecewise functions, always check the value at the break point to see if the function is continuous or if there is a jump.

Sketch a graph of f(x) = { x - 5, if x < 1; -4 + 2x, if x ≥ 1}.

Learn how to sketch a graph of the piecewise function f(x) = {x - 5 for x < 1, -4 + 2x for x ≥ 1}, including how to plot lines with open and closed circles at break points. Suitable for students in Grades 9-12.

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